A System of Nonlinear Partial Differential Equations Describing Cylindrical Plasma Collapse

摘要

We consider the snow-plough model for describing cylindrical plasma collapse for a specified constant driving term. This coupled nonlinear system consists of five partial differential equations in two independ¬ent variables, one of which is the time variable. Generally, the initial value problem for similar systems is improperly posed. However, here we show that by direct construction of the unique solution, explicitly in terms of the initial data, the solution exists for all positive times and is generally an infinitely differentiable function of the independent variables. Nevertheless, the solution always develops a nonphysical singularity after a certain positive time, and thereafter ceases to describe the underlying physical situa¬tion. Our theory leads to an a priori bound in terms of the initial data, on the time interval during which the snow-plough model is physically realistic. We discuss several examples which illustrate the pathol¬ogies exhibited by the solution.